2. Write about confidence region. 5marks

Ans. Confidence region is a multi-dimensional generalization of a confidence interval. It
is a set of points in an n-dimensional space, often represented as an ellipsoid around a
point which is an estimated solution to a problem, although other shapes can occur.

• Confidence regions are multivariate extensions of univariate confidence intervals.
• These regions usually cover the complete range of the data that went into the

model, and incorporate both uncertainly in the parameter estimates and prediction
error.

• Confidence regions are sometimes called as INTERFERENCE REGIONS, including that
these are regions where one infers something about the likelihood of the
parameters existing.

• For the point estimation, a single number is calculated, called the point estimator.
Instead, it is often more desirable to compute an interval of values that is likely to
contain the true value of the parameter. Because the variability of sample to sample,
we can never say for sure if the interval contains the parameter. However, we would
like to say that the proposed interval will contain the true value with a specified high
probability. This probability is called the CONFIDENCE and the INTERVAL is typically
taken as 90%, 95%, 99%. For any confidence level, the corresponding interval is
computed as:
CI =( mean – EM, mean +EM) Or P% CONFIDENCE INTERVAL =±EM

• Confidence region can be defined for any probability distribution.
• The experimenter can choose the significance level and the shape of the region, and

then the size of the region is determined by the probability distribution. A natural
choice is to use as a boundary a set of points with constant CHI-SQUARE VALUE.

• One approach is to use a linear approximation to the non-linear model, which may
be a close approximation in the vicinity of the solution, and then apply the analysis
for a linear problem to find an approximate confidence region. This may be a
reasonable approach if the confidence region is not very large.

• Another approach considers that the regions of equivalent parameter values must
enclose parameter for which the loss function is nearly the same or at any rate less
different than some threshold.

• Interpretation of confidence region:
i. It gives a range of possible value for the parameter.

ii. It provides a measure of the extent to which a simple estimate is likely to
differ from the true population value.

iii. It indicates with a stated level of certainty, the range of values within which
the true population mean is likely to lie.

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iv. It gives information about the closeness of the sample to unknown
population parameter.

v. Confidence Interval depends on the level of confidence.

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